Magnetic Schr\"odinger operators on periodic discrete graphs

We consider magnetic Schr"odinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of non-degenerate bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We estimate the Lebesgue measure of the spectrum in terms of the Betti numbers and show that these estimates become identities for specific graphs. We estimate a variation of the spectrum of the Schr"odinger operators under a perturbation by a magnetic field in terms of magnetic fluxes. The proof is based on Floquet theory and a precise representation of fiber magnetic Schr"odinger operators constructed in the paper.